Question

L, C and R represent the physical quantities inductance, capacitance and resistance respectively. Which of the following combinations have dimensions of frequency?
(a) $\frac{1}{RC}$
(b) $\frac{R}{L}$
(c) $\frac{1}{\sqrt{LC}}$
(d) C/L.

Answer 1

(a) $\frac{1}{RC}$

(b) $\frac{R}{L}$

(c) $\frac{1}{\sqrt{LC}}$

The time constant of the RC circuit is given by
$\tau =RC$
On taking the reciprocal of the above relation, we get
$f_1=\frac{1}{RC}$ ...(1)
f₁ will have the dimensions of the frequency.
The time constant of the LR circuit is given by
$\tau =\frac{L}{R}$
On taking the reciprocal of the above relation, we get
$f_2=\frac{R}{L}$ ...(2)
f₂ will have the dimensions of the frequency.
On multiplying eq. (1) and (2), we get
$$\begin{gathered} f_1{f}_2=\frac{1}{LC} \\ \Rightarrow \sqrt{f_1{f}_2}=\frac{1}{\sqrt{LC}} \end{gathered}$$
Thus, $\sqrt{f_1{f}_2}$ will have the dimensions of the frequency.